The maximum amount of money that a person is willing to spend on a kind of product, for example, a camera or a car, is called the person's reservation price. The person's reservation price can also be represented as a probability distribution (the probability of a person buying a product at a price). When a product is sold at a single price, the customers with a lower reservation price will not buy the product and the customers with a higher reservation price are not fully exploited.
A demand curve is typically used to represent the relationship between the number of units sold and the price of a product for potential buyers of the product. FIG. 1 (Prior Art) shows an exemplary graph 1 that illustrates a demand curve 2. The number of units sold is indicated on the vertical axis and price is indicated on the horizontal axis. Demand curve 2 shows that the number of units sold decreases as price increases.
The total area under demand curve 2 is the maximum amount of money (revenue) that can be made through selling the product if each customer is fully exploited at his reservation price. However, fully exploiting each customer at his reservation price is impractical as too many different prices must be used. Therefore, in practice, a single price is often chosen. FIG. 1 (Prior Art) shows an exemplary single selling price p. At price p, the number of units sold is n. The area within rectangle 3 represents the revenue that will be obtained from selling n units at price p. It can be seen that the area within rectangle 3 is significantly less than the total area under demand curve 2.
In some instances several prices are used for selling the same product. These different prices are usually supported by differences in the details of functionalities and services that the customer will get. This is called multiple discriminant pricing or product price differentiation. By having multiple prices, a larger area underneath the demand curve is obtained, giving higher revenue. In other words, customers are segmented according to the price that they will pay for the product.
When more than one price is to be used for selling a product, a decision must be made as to how many prices are to be used. Too many different prices on a single product are not practical. However, too few prices for a product will not exploit the potential revenue indicated by the demand curve.
Once the number of prices to be used is determined the prices must be chosen. In practice, prices are typically chosen according to factors independent of the demand curve for the product. Factors that are commonly used include prior experience for similar products, prices for competing products, etc. However, often the chosen prices do not result in maximization of revenue.
Academic exercises are sometimes taught in which simplistic models of a demand curve are studied for the purpose of maximizing revenue. When the demand curve is a straight line or some other simple shape (e.g., a uniform curvature), these models give reasonable results. However, in real life situations, demand curves are complex shapes that are not uniform or simplistic. Typically, demand curves for products have arbitrary shapes that tend to be concave over a large range of prices. These simplistic models do not work on demand curves having arbitrary shapes, even when the arbitrary shapes are concave. Therefore, in real life situations, the techniques of these academic processes are ineffective for determining prices that maximize revenue.
Even more complex are multiple product scenarios. In multiple product scenarios, prices must be determined for several products, each having its own demand curve. Because of the complexity in multiple-product scenarios, prices are often chosen independently for each product, not taking into account the other products during the decision process. This does not result in maximization of revenue for the entire group of products.
Accordingly, there is a need for a method and apparatus that will determine prices that maximize revenue for a given product having a demand curve. Also, there is a need for a method and apparatus that will determine either a single price or multiple prices that maximize revenue for a demand curve having an arbitrary shape that is concave. In addition, a method and apparatus is needed that meets the above need and that can determine prices that maximize revenue in a multiple product environment.